【作者】 李荣军；

【导师】 周叔子；

【作者基本信息】 湖南大学 ， 计算数学， 2004， 硕士

【摘要】 瀑布型多重网格法是求解大型边值问题的一种有效迭代解法。其主要的优点是不要求粗网格校正，故又称单步多重网格法。Gisela Timmermann用瀑布型多重网格法对半线性椭圆问题进行了求解，在粗网格上用牛顿法(Newton)将由线性有限元离散而得到的非线性方程组线性化，在细网格上用瀑布型多重网格法解这个牛顿方程，并且提出了算法和对算法的收敛性进行了研究。 本文将瀑布型多重网格法推广到半线性抛物问题，证明在能量范数下算法误差的最优阶，且有最优或有拟最优的计算工作量，并进行数值试验。 本文以半线性二阶抛物型偏微分方程初边值问题为模型问题构造了瀑布型多重网格法(CMG)，在最粗网格上用牛顿法(Newton)将由线性有限元离散而得到的非线性方程组线性化，在细网格上用瀑布型多重网格法解这个牛顿方程首先采用Richardson迭代法作为光滑子，我们证明了瀑布型多重网格法对二维半线性抛物型边值问题在能量范数下可获得最优收敛阶。然后又对采用共轭梯度法(CG)作为光滑子进行了研究，并得到了在此情况下，瀑布型多重网格法对二维半线性抛物型边值问题在能量范数下可获得最优收敛阶。同时对这两种情形，分析了计算工作量，得到了工作量的最优性或拟最优性，这说明在半线性情形的计算工作量与线性情形是大致相当的。数值实验也显示了该算法的有效性。更多还原

【Abstract】 The cascadic multigrid method has been shown to be one of the most efficient iterative techniques for solving large scale boundary value problems.The main advantage of the method is coarse-grid-correction free,and as a result it can be viewed as a one-way multigrid method. Gisela Timmermann proposed a cascadic multigrid for a semilinear elliptic problem. On the coarest grid the nonlinear equations arising from linear finite element discretizations are solved by Newton’s method.On the fine grid the Newton’s equations is solved by the cascadic multigrid method.In this paper we extend the cascadic multigrid method to semilinear parabolic problems. It has been proved that the method has optimal convergence order of the error in energy norm, and has the optimal or quasi-optimal computation complexity.We construct cascadic multigrid method for a model problem-a semilinear parabolic problem. On the coarest grid the nonlinear equations arising from linear finite element discretizations are solved by Newton’s method.On the fine grid the Newton’s equations is solved by the cascadic multigrid method.At first we use Richardson iteration as smoothing operator and prove the method has optimal convergence order for the error in the energy norm to 2-D semilinear parabolic problem. Then we use conjugate gradient(CG) as smoothers and prove this method has optimal convergence order of the error in the energy norm also. For these two cases, we analyse the computational complexity.The optimality or quasi-optimalityof the computation of the method is shown.This fact displays that for cascadic multigrid the computational work in semilinear case is almost the same as that in linear case.Finally a numerical experiment is given to the effectiveness of the method.更多还原